Mathew Barzal scored for the Islanders (15-15-6), who had won four of five. Semyon Varlamov made 27 saves.
New York is 2-13-2 and has been outscored 58-27 against teams currently in position to qualify for the Stanley Cup Playoffs.
"L.A. had nothing tonight, they really didn't have much, and I thought we generated a little bit more," Islanders coach Barry Trotz said. "They defended well, they clogged it up. They defended well, and it was hard to generate. They were looking for a low-scoring game, and we generally do. We're not going to trade chances with some of the teams, but we've got to go to the net. We've got to put the puck in play."
Byfield, who was playing in his 10th NHL game, and fourth this season, gave the Kings a 1-0 lead with a wrist shot from the between the face-off circles at 17:05 of the first period.
"I knew I had the one defender beat and wanted to take it kind of wide, almost like freeze the defender. … I got another step and just a quick release," Byfield said. "It found the back of the net. It was a good feeling."
Athanasiou made it 2-0 with six seconds left in the second period when his wrist shot from along the left boards went in off the crossbar after it appeared to catch Varlamov by surprise.
It was Athanasiou's first game since Dec. 14; he missed 16 games because of a lower-body injury.
"That was a big goal," Kings defenseman Drew Doughty said. "That put us up two. [Athanasiou] hasn't played in a long, long time, and to see him score, [he's] so skilled and I think he's scored 30 in the League before (2018-19 with Detroit Red Wings), so we need him in the lineup and we need him to be playing to be at our best. That was huge for us."